Relative Milnor $K$-groups and differential forms of split nilpotent extensions

TL;DR

This paper proves a canonical isomorphism between relative Milnor K-groups and differential forms for split nilpotent extensions of rings, under certain invertibility and stability conditions.

Contribution

It establishes a new isomorphism linking Milnor K-theory and differential forms in the context of split nilpotent extensions, expanding understanding of algebraic K-theory.

Findings

Isomorphism between $K^{M}_{n+1}(R,I)$ and $rac{ ext{Omega}^n_{R,I}}{d ext{Omega}^{n-1}_{R,I}}$

Requires $N!$ to be invertible in $R$

Assumes $R$ is weakly 5-fold stable

Abstract

Let $R$ be a commutative ring and $I\subset R$ be a nilpotent ideal such that the quotient $R/I$ splits out of $R$ as a ring. Let $N$ be a natural number such that ${I^N=0}$. We establish a canonical isomorphism between the relative Milnor $K$-group $K^{M}_{n+1}(R,I)$ and the quotient of the relative module of differential forms $\Omega^n_{R,I}/d\,\Omega^{n-1}_{R,I}$ assuming that $N!$ is invertible in $R$ and that the ring $R$ is weakly $5$-fold stable. The latter means that any $4$-tuple of elements in $R$ can be shifted by an invertible element to become a $4$-tuple of invertible elements.